Nanocolloidal hydrogel mimics the structure and nonlinear mechanical properties of biological fibrous networks.

Fibrous networks formed by biological polymers such as collagen or fibrin exhibit nonlinear mechanical behavior. They undergo strong stiffening in response to weak shear and elongational strains, but soften under compressional strain, in striking difference with the response to the deformation of flexible-strand networks formed by molecules. The nonlinear properties of fibrous networks are attributed to the mechanical asymmetry of the constituent filaments, for which a stretching modulus is significantly larger than the bending modulus. Studies of the nonlinear mechanical behavior are generally performed on hydrogels formed by biological polymers, which offers limited control over network architecture. Here, we report an engineered covalently cross-linked nanofibrillar hydrogel derived from cellulose nanocrystals and gelatin. The variation in hydrogel composition provided a broad-range change in its shear modulus. The hydrogel exhibited both shear-stiffening and compression-induced softening, in agreement with the predictions of the affine model. The threshold nonlinear stress and strain were universal for the hydrogels with different compositions, which suggested that nonlinear mechanical properties are general for networks formed by rigid filaments. The experimental results were in agreement with an affine model describing deformation of the network formed by rigid filaments. Our results lend insight into the structural features that govern the nonlinear biomechanics of fibrous networks and provide a platform for future studies of the biological impact of nonlinear mechanical properties.

How a Chain Can Be Extended While Its Bonds Are Compressed.

Extending polymer chains results in a positive chain tension, fch, primarily due to conformational restrictions. At the level of individual bonds, however, tension, fb, is either negative or positive and depends on both chain tension and bulk pressure. Typically, the chain and bond tension are assumed to be directly related. In specific systems, however, this dependence may not be intuitive, whereby fch increases while fb decreases; i.e., the entire chain is extended while bonds are compressed. Specifically, increasing the grafting density of a polymer brush results in chain extension along the direction perpendicular to the grafting surface while the underlying bonds are compressed. Similarly, upon compression of polymer networks, the extension of chains oriented in the "free" direction increases while their bonds are getting more compressed. We demonstrate this phenomenon in molecular dynamics simulations and explain it by the fact that the pressure contribution to fb is dominant over a wide range of network deformations and brush grafting densities.

Elasticity of Slide-Ring Gels.

Slide-ring gels are polymer networks with cross-links that can slide along the chains. In contrast to conventional unentangled networks with cross-links fixed along the chains, the slide-ring networks are strain-softening and distribute tension much more uniformly between their strands due to the so-called "pulley effect". The sliding of cross-links also reduces the elastic modulus in comparison with the modulus of conventional networks with the same number density of cross-links and elastic strands. We develop a single-chain model to account for the redistribution of monomers between network strands of a primary chain. This model takes into account both the pulley effect and fluctuations in the number of monomers per network strand. The pulley effect leads to modulus reduction and uniform tension redistribution between network strands, while fluctuations in the number of strand monomers dominate the strain-softening, the magnitude of which decreases upon network swelling and increases upon deswelling.

Fibrous hydrogels under biaxial confinement.

Confinement of fibrous hydrogels in narrow capillaries is of great importance in biological and biomedical systems. Stretching and uniaxial compression of fibrous hydrogels have been extensively studied; however, their response to biaxial confinement in capillaries remains unexplored. Here, we show experimentally and theoretically that due to the asymmetry in the mechanical properties of the constituent filaments that are soft upon compression and stiff upon extension, filamentous gels respond to confinement in a qualitatively different manner than flexible-strand gels. Under strong confinement, fibrous gels exhibit a weak elongation and an asymptotic decrease to zero of their biaxial Poisson's ratio, which results in strong gel densification and a weak flux of liquid through the gel. These results shed light on the resistance of strained occlusive clots to lysis with therapeutic agents and stimulate the development of effective endovascular plugs from gels with fibrous structures for stopping vascular bleeding or suppressing blood supply to tumors.

Theory of Flexible Polymer Networks: Elasticity and Heterogeneities.

A review of the main elasticity models of flexible polymer networks is presented. Classical models of phantom networks suggest that the networks have a tree-like structure. The conformations of their strands are described by the model of a combined chain, which consists of the network strand and two virtual chains attached to its ends. The distribution of lengths of virtual chains in real polydisperse networks is calculated using the results of the presented replica model of polymer networks. This model describes actual networks having strongly overlapping and interconnected loops of finite sizes. The conformations of their strands are characterized by the generalized combined chain model. The model of a sliding tube is represented, which describes the general anisotropic deformations of an entangled network in the melt. I propose a generalization of this model to describe the crossover between the entangled and phantom regimes of a swollen network. The obtained dependence of the Mooney-Rivlin parameters C 1 and C 2 on the polymer volume fraction is in agreement with experiments. The main results of the theory of heterogeneities in polymer networks are also discussed.

Ion Pairing and the Structure of Gel Coacervates.

A scaling model for the structure of coacervates is presented for mixtures of oppositely-charged polyelectrolytes of both symmetric and asymmetric charge-densities for different degrees of electrostatic strength and levels of added salt. At low electrostatic strengths, weak coacervates, with the energy of electrostatic interactions between charges less than the thermal energy, k B T, are liquid. At higher electrostatic strengths, strong coacervates are gels with crosslinks formed by ion pairs of opposite charges bound to each other with energy higher than k B T. Charge-symmetric coacervates are formed for mixtures of oppositely-charged polyelectrolytes with equal and opposite charge-densities. While charge-symmetric weak coacervates form a semidilute polymer solution with a correlation length equal to the electrostatic blob size, charge-symmetric strong coacervates form reversible gels with a correlation length on the order of the distance between bound ion pairs. Charge-asymmetric coacervates are formed from mixtures of oppositely-charged polyelectrolytes with different charge-densities. While charge-asymmetric weak coacervates form double solutions with two correlation lengths and qualitatively different chain conformations of polycations and polyanions, charge-asymmetric strong coacervates form bottlebrush and star-like gels. Unlike liquid coacervates, for which an increase in the concentration of added salt screens electrostatic interactions, causing structural rearrangement and eventually leads to their dissolution, the salt does not affect the structure of strong coacervates until ion pairs dissociate and the gel disperses.

Structure of Liquid Coacervates formed by Oppositely Charged Polyelectrolytes.

We develop a scaling theory and perform molecular dynamic simulations of weakly interacting coacervates with electrostatic interaction energy per charge less than thermal energy kT. Such liquid coacervates formed by oppositely charged polyelectrolytes can be asymmetric in charge density and number of charges per chain. We predict that these coacervates form interpenetrating solutions with two correlation lengths and two qualitatively different types of conformations of polyelectrolytes with lower and higher charge densities, which are analogous to chain conformations in quasi-neutral and in polyelectrolyte solutions, respectively. Weaker charged chains are attracted to and adsorbed on stronger charged chains forming a screening "coat" around the stronger charged polyelectrolytes. Salt added at lower concentrations screens the repulsion between stronger charged chains, thereby reducing the thickness of the screening coat and resulting in the non-zero net polymer charge in the coacervate. At higher salt concentrations salt screens the attraction between oppositely charged chains, decreasing the coacervate concentration and its polymeric charge density. Thus, we predict a non-monotonic salt concentration dependence of polymeric charge density for asymmetric coacervates. Phase diagram for a mixture of oppositely charged polyelectrolytes at various compositions is proposed for different salt concentrations.

Enhanced nanochannel translocation and localization of genomic DNA molecules using three-dimensional nanofunnels.

The ability to precisely control the transport of single DNA molecules through a nanoscale channel is critical to DNA sequencing and mapping technologies that are currently under development. Here we show how the electrokinetically driven introduction of DNA molecules into a nanochannel is facilitated by incorporating a three-dimensional nanofunnel at the nanochannel entrance. Individual DNA molecules are imaged as they attempt to overcome the entropic barrier to nanochannel entry through nanofunnels with various shapes. Theoretical modeling of this behavior reveals the pushing and pulling forces that result in up to a 30-fold reduction in the threshold electric field needed to initiate nanochannel entry. In some cases, DNA molecules are stably trapped and axially positioned within a nanofunnel at sub-threshold electric field strengths, suggesting the utility of nanofunnels as force spectroscopy tools. These applications illustrate the benefit of finely tuning nanoscale conduit geometries, which can be designed using the theoretical model developed here.Forcing a DNA molecule into a nanoscale channel requires overcoming the free energy barrier associated with confinement. Here, the authors show that DNA injected through a funnel-shaped entrance more efficiently enters the nanochannel, thanks to facilitating forces generated by the nanofunnel geometry.

Self-Similar Conformations and Dynamics in Entangled Melts and Solutions of Nonconcatenated Ring Polymers.

A scaling model of self-similar conformations and dynamics of nonconcatenated entangled ring polymers is developed. Topological constraints force these ring polymers into compact conformations with fractal dimension df = 3 that we call fractal loopy globules (FLGs). This result is based on the conjecture that the overlap parameter of subsections of rings on all length scales is the same and equal to the Kavassalis-Noolandi number OKN ≈ 10-20. The dynamics of entangled rings is self-similar and proceeds as loops of increasing sizes are rearranged progressively at their respective diffusion times. The topological constraints associated with smaller rearranged loops affect the dynamics of larger loops through increasing the effective friction coefficient but have no influence on the entanglement tubes confining larger loops. As a result, the tube diameter defined as the average spacing between relevant topological constraints increases with time t, leading to "tube dilation". Analysis of the primitive paths in molecular dynamics simulations suggests a complete tube dilation with the tube diameter on the order of the time-dependent characteristic loop size. A characteristic loop at time t is defined as a ring section that has diffused a distance equal to its size during time t. We derive dynamic scaling exponents in terms of fractal dimensions of an entangled ring and the underlying primitive path and a parameter characterizing the extent of tube dilation. The results reproduce the predictions of different dynamic models of a single nonconcatenated entangled ring. We demonstrate that traditional generalization of single-ring models to multi-ring dynamics is not self-consistent and develop a FLG model with self-consistent multi-ring dynamics and complete tube dilation. This selfconsistent FLG model predicts that the longest relaxation time of nonconcatenated entangled ring polymers scales with their degree of polymerization N as τrelax ~ N7/3, while the diffusion coefficient of these rings scales as D3d ~ N-5/3. For the entangled solutions and melts of rings, we predict power law stress relaxation function G(t) ~ t-3/7 at t < τrelax without a rubbery plateau and the corresponding viscosity scaling with the degree of polymerization N as η ~ N4/3. These theoretical predictions are in good agreement with recent computer simulations and are consistent with experiments of melts of nonconcatenated entangled rings.

Molecular structure of bottlebrush polymers in melts.

Bottlebrushes are fascinating macromolecules that display an intriguing combination of molecular and particulate features having vital implications in both living and synthetic systems, such as cartilage and ultrasoft elastomers. However, the progress in practical applications is impeded by the lack of knowledge about the hierarchic organization of both individual bottlebrushes and their assemblies. We delineate fundamental correlations between molecular architecture, mesoscopic conformation, and macroscopic properties of polymer melts. Numerical simulations corroborate theoretical predictions for the effect of grafting density and side-chain length on the dimensions and rigidity of bottlebrushes, which effectively behave as a melt of flexible filaments. These findings provide quantitative guidelines for the design of novel materials that allow architectural tuning of their properties in a broad range without changing chemical composition.

Universal behavior of hydrogels confined to narrow capillaries.

Flow of soft matter objects through one-dimensional environments is important in industrial, biological and biomedical systems. Establishing the underlying principles of the behavior of soft matter in confinement can shed light on its performance in many man-made and biological systems. Here, we report an experimental and theoretical study of translocation of micrometer-size hydrogels (microgels) through microfluidic channels with a diameter smaller than an unperturbed microgel size. For microgels with different dimensions and mechanical properties, under a range of applied pressures, we established the universal principles of microgel entrance and passage through microchannels with different geometries, as well as the reduction in microgel volume in confinement. We also show a non-monotonic change in the flow rate of liquid through the constrained microgel, governed by its progressive confinement. The experimental results were in agreement with the theory developed for non-linear biaxial deformation of unentangled polymer gels. Our work has implications for a broad range of phenomena, including occlusion of blood vessels by thrombi and needle-assisted hydrogel injection in tissue engineering.

Hopping Diffusion of Nanoparticles in Polymer Matrices.

We propose a hopping mechanism for diffusion of large nonsticky nanoparticles subjected to topological constraints in both unentangled and entangled polymer solids (networks and gels) and entangled polymer liquids (melts and solutions). Probe particles with size larger than the mesh size ax of unentangled polymer networks or tube diameter ae of entangled polymer liquids are trapped by the network or entanglement cells. At long time scales, however, these particles can diffuse by overcoming free energy barrier between neighboring confinement cells. The terminal particle diffusion coefficient dominated by this hopping diffusion is appreciable for particles with size moderately larger than the network mesh size ax or tube diameter ae . Much larger particles in polymer solids will be permanently trapped by local network cells, whereas they can still move in polymer liquids by waiting for entanglement cells to rearrange on the relaxation time scales of these liquids. Hopping diffusion in entangled polymer liquids and networks has a weaker dependence on particle size than that in unentangled networks as entanglements can slide along chains under polymer deformation. The proposed novel hopping model enables understanding the motion of large nanoparticles in polymeric nanocomposites and the transport of nano drug carriers in complex biological gels such as mucus.

Mobility of Nonsticky Nanoparticles in Polymer Liquids.

We use scaling theory to derive the time dependence of the mean-square displacement 〈Δr2〉 of a spherical probe particle of size d experiencing thermal motion in polymer solutions and melts. Particles with size smaller than solution correlation length ξ undergo ordinary diffusion (〈Δr2 (t)〉 ~ t) with diffusion coefficient similar to that in pure solvent. The motion of particles of intermediate size (ξ < d < a), where a is the tube diameter for entangled polymer liquids, is sub-diffusive (〈Δr2 (t)〉 ~ t1/2) at short time scales since their motion is affected by sub-sections of polymer chains. At long time scales the motion of these particles is diffusive and their diffusion coefficient is determined by the effective viscosity of a polymer liquid with chains of size comparable to the particle diameter d. The motion of particles larger than the tube diameter a at time scales shorter than the relaxation time τ e of an entanglement strand is similar to the motion of particles of intermediate size. At longer time scales (t > τ e ) large particles (d > a) are trapped by entanglement mesh and to move further they have to wait for the surrounding polymer chains to relax at the reptation time scale τrep. At longer times t > τrep, the motion of such large particles (d > a) is diffusive with diffusion coefficient determined by the bulk viscosity of the entangled polymer liquids. Our predictions are in agreement with the results of experiments and computer simulations.

Bond Tension in Tethered Macromolecules.

The paper presents scaling analysis of mechanical tension generated in densely branched macromolecules tethered to a solid substrate with a short linker. Steric repulsion between branches results in z-fold amplification of tension in the linker, where z is the number of chain-like arms. At large z ~ 100-1000, the generated tension may exceed the strength of covalent bonds and sever the linker. Two types of molecular architectures were considered: polymer stars and polymer "bottlebrushes" tethered to a solid substrate. Depending on the grafting density, one distinguishes the so-called mushroom, loose grafting, and dense grafting regimes. In isolated (mushroom) and loosely tethered bottlebrushes, the linker tension is by a factor of [Formula: see text] smaller than the tension in a tethered star with the same number of arms z. In densely tethered stars, the effect of interchain distance (d) and number of arms (z) on the magnitude of linker tension is given by f ≅ f0z3/2(b/d) for stars in a solvent environment and f ≅ f0z2 (b/d)2 for dry stars, where b is the Kuhn length and f0 ≅ kBT/b is intrinsic bond tension. These relations are also valid for tethered bottlebrushes with long side chains. However, unlike molecular stars, bottlebrushes demonstrate variation of tension along the backbone f ≅ f0s z1/2 / d as a function of distance s from the free end of the backbone. In dense brushes [Formula: see text] with z ≅ 1000, the backbone tension increases from f ≅ f0 = 1 pN at the free end of the backbone (s ≅ b) to its maximum f ≅ zf0 ≅ 1 nN at the linker to the substrate (s ≅ zb).

Tension amplification in molecular brushes in solutions and on substrates.

Molecular bottle-brushes are highly branched macromolecules with side chains densely grafted to a long polymer backbone. The brush-like architecture allows focusing of the side-chain tension to the backbone and its amplification from the pico-Newton to nano-Newton range. The backbone tension depends on the overall molecular conformation and the surrounding environment. Here we study the relation between the tension and conformation of the molecular brushes in solutions, melts, and on substrates. In solutions, we find that the backbone tension in dense brushes with side chains attached to every backbone monomer is on the order of f(0)N(3/8) in athermal solvents, f(0)N(1/3) in theta solvents, and f(0) in poor solvents and melts, where N is the degree of polymerization of side chains, f(0) approximately equal k(B)T/b is the maximum tension in side chains, b is the Kuhn length, k(B) is Boltzmann's constant, and T is the absolute temperature. Depending on the side chain length and solvent quality, molecular brushes develop tension on the order of 10-100 pN, which is sufficient to break hydrogen bonds. Significant amplification of tension occurs upon adsorption of brushes onto a substrate. On a strongly attractive substrate, maximum tension in the brush backbone is approximately f(0)N, reaching values on the order of several nano-Newtons, which exceeds the strength of a typical covalent bond. At low grafting density and high spreading parameter, the cross-sectional profile of an adsorbed molecular brush is approximately rectangular with a thickness approximately b (A/S)1/2, where A is the Hamaker constant, and S is the spreading parameter. At a very high spreading parameter (S > A), the brush thickness saturates at monolayer approximately b. At a low spreading parameter, the cross-sectional profile of adsorbed molecular brush has a triangular tent-like shape. In the cross-over between these two opposite cases, covering a wide range of parameter space, the adsorbed molecular brush consists of two layers. Side chains in the lower layer gain surface energy due to the direct interaction with the substrate, while the second layer spreads on the top of the first layer. Scaling theory predicts that this second layer has a triangular cross-section with width R approximately N(3/5) and height h approximately N(2/5). Using self-consistent field theory we calculate the cap profile y(x) = h(1 - x2/R2)2, where x is the transverse distance from the backbone. The predicted cap shape is in excellent agreement with both computer simulation and experiment.

Amplification of tension in branched macromolecules.

We propose a method for the design of branched macromolecules that are capable of building up high tension ( approximately nN) in their covalent bonds without applying an external force. The tension is self-generated due to repulsion between branches and depends on the molecular architecture leading to amplification and focusing of tension in specific bonds. The simplest architecture is a pom-pom composed of a linear spacer and two z-arm stars at its ends resulting in z-fold tension amplification. Adsorption of those macromolecules on a substrate results in further increase in tension as compared to molecules in solution.

Metastable lattice of droplets in phase separating polymer blends.

Phase separation in a polymer mixture with an off-critical composition is described by a Ginzburg-Landau Hamiltonian that contains both cubic and quartic terms in the deviation of composition from its mean value in the homogeneous phase. Our analysis suggests that when a blend is brought in the vicinity of the spinodal, the initial homogeneous phase becomes unstable against the formation of a metastable lattice of spherical droplets whose lifetime diverges in the limit of infinite molecular weight. The composition of the droplets approaches that of the background phase and their size diverges with the approach to the critical point, but the composition contrast is enhanced and droplet radii become comparable to polymer dimensions, away from criticality. The connection between our predictions and the results of recent neutron scattering experiments is discussed, and new experiments that could probe the proposed droplet lattice are proposed.